Sharp logarithmic inequalities for Riesz transforms

Let d be a given positive integer and let {Rj }d j=1 denote the collection of Riesz transforms on Rd. For anyK >2/π we determine the optimal constant L such that the following holds. For any locally integrable Borel function f on Rd , any Borel subset A of Rd and any j = 1, 2, . . . , d we have A Rjf (x) dx K Rd Ψ f (x) dx + |A| · L. Here Ψ(t) = (t +1) log(t +1)−t for t 0. The proof is based on probabilistic techniques and the existence of certain special harmonic functions. As a by-product, we obtain related sharp estimates for the so-called re-expansion operator, an important object in some problems of mathematical physics. © 2012 Elsevier Inc. All rights reserved.